1. Field of the Invention
The present invention relates to an apparatus for and a method of measuring a jitter in a microcomputer. More particularly, the present invention relates to an apparatus for and a method of measuring a jitter in a clock generating circuit used in a microcomputer.
2. Description of the Related Art
In the past thirty years, the number of transistors on a VLSI (very large scale integrated circuit) chip has been exponentially increasing in accordance with Moore""s law, and the clock frequency of a microcomputer has also been exponentially increasing in accordance with Moore""s law. At present time, the clock frequency is about to exceed the limit of 1.0 GHz. (For example, see: Naoaki Aoki, H. P. Hofstee, and S. Dong; xe2x80x9cGHz MICROPROCESSORxe2x80x9d, INFORMATION PROCESSING vol. 39, No. 7, July 1998.) FIG. 1 is a graph showing a progress of clock period in a microcomputer disclosed in Semiconductor Industry Association: xe2x80x9cThe National Technology Roadmap for Semiconductors, 1997xe2x80x9d. In FIG. 1, an RMS jitter (root-mean-square jitter) is also plotted.
In a communication system, a carrier frequency and a carrier phase, or symbol timing are regenerated by applying non-linear operations to a received signal and by inputting the result of the non-linear process to a phase-locked loop (PLL) circuit. This regeneration corresponds to the maximum likelihood parameter estimation. However, when a carrier or a data cannot correctly be regenerated from the received signal due to an influence of a noise or the like, a retransmission can be requested to the transmitter. In a communication system, a clock generator is formed on a separate chip from the other components. This clock generator is formed on a VLSI chip using a.bipolar technology, GaAs technology or a CMOS technology.
In each of many microcomputers, an instruction execution is controlled by a clock signal having a constant period. The clock period of this clock signal corresponds to a cycle time of a microcomputer. (For example, see: Mike Johnson; xe2x80x9cSuperscale Microprocessor Designxe2x80x9d, Prentice-Hall, Inc., 1991.) If the clock period is too short, a synchronous operation becomes impossible and the system is locked. In a microcomputer, a clock generator is integrated in a same chip where other logical circuits are integrated. FIG. 2 shows, as an example, a Pentium chip. In FIG. 2, a white square (xe2x96xa1) indicates a clock generating circuit. This microcomputer is produced utilizing a CMOS (complementary metal-oxide semiconductor) processing.
In a communication system, the average jitter or the RMS jitter is important. The RMS jitter contributes to an average noise of signal-to-noise ratio and increases the bit error rate. On the other hand, in a microcomputer, the worst instantaneous value of some parameter determines the operation frequency. That is, the peak-to-peak jitter (the worst value of jitter) determines the upper limit of the operation frequency.
Therefore, for testing of a PLL circuit in a microcomputer, there is required a method of measuring an instantaneous value of jitter accurately and in a short period of time. However, since a measurement of a jitter has been developed in the area of communications, there is no measuring method, in the present state, corresponding to this requirement in the area of microcomputers. It is an object of the present invention to provide a method of measuring an instantaneous value of jitter accurately and in a short period of time.
On the contrary, for testing of a PLL circuit in a communication system, there is required a method of measuring an RMS jitter accurately. Although it takes approximately 10 minutes of measuring time, a measuring method actually exists and is practically used. FIG. 3 collectively shows comparisons of clock generators between a microcomputer and a communication system.
A phase-locked loop circuit (PLL circuit) is a feedback system. In a PLL circuit, a frequency and a phase xcex8i of a given reference signal are compared with a frequency and a phase xcex80 of an internal signal source, respectively to control the internal signal source, using the differences therebetween, such that the frequency difference or the phase difference can be minimized. Therefore, a voltage-controlled oscillator (VCO) which is an internal signal source of a PLL circuit comprises a component or components the delay time of which can be varied. When a DC voltage is inputted to this oscillator, a repetitive waveform having a constant period proportional to the direct current value is outputted.
The PLL circuit relating to the present invention comprises a phase-frequency detector, a charge pump circuit, a loop filter and a VCO. FIG. 4 shows a basic circuit configuration of a PLL circuit in a block diagram form. Next, the operation of each of the circuit components will be briefly described.
A phase-frequency detector is a digital sequential circuit. FIG. 5 is a block diagram showing a circuit configuration of a phase-frequency detector comprising two D-type flip-flops D-FF1 and D-FF2 and an AND gate. A reference clock is applied to a clock terminal ck of the first D-type flip-flop D-FF1, and a PLL clock is applied to a clock terminal ck of the second D-type flip-flop D-FF2. A logical value xe2x80x9c1xe2x80x9d is supplied to each data input terminal D.
In the circuit configuration described above, when each of the two Q outputs of the both flip-flops becomes xe2x80x9c1xe2x80x9d at the same time, the AND gate resets the both flip-flops. The phase-frequency detector outputs, depending on the phase difference and the frequency difference between the two input signals, an UP signal for increasing the frequency or a DOWN signal for decreasing the frequency. (For example, see: R. Jacob Baker, Henry W. Li, and David E. Boyce; xe2x80x9cCMOS Circuit Design, Layout, and Simulationxe2x80x9d, IEEE Press, 1998.)
FIG. 6 shows a state transition diagram of a phase-frequency detector (PFD). The phase-frequency detector transits the state by rise edges of a reference clock and a PLL clock. For example, as shown in FIG. 7, when the frequency of a reference clock is 40 MHz and the frequency of a PLL clock is 37 MHz, in order to increase the frequency, an UP signal is outputted during a time interval between the two rise edges. A similar operation is also performed when a phase difference is present between the reference clock and the PLL clock. The phase-frequency detector has the following characteristics compared with a phase detector using an Exclusive OR circuit. (For example, see: R. Jacob Baker, Henry W. Li, and David E. Boyce; xe2x80x9cCMOS Circuit Design, Layout, and Simulationxe2x80x9d, IEEE Press, 1998.)
(i) The phase-frequency detector operates at a rising edge of an input clock, and does not relate to the shape of the waveform such as a pulse width of the clock.
(ii) The phase-frequency detector is not locked by a harmonic of the reference frequency.
(iii) Since both of the two outputs are logical xe2x80x9c0xe2x80x9d during a time period when the loop is locked, a ripple is not generated at the output of the loop filter.
The phase-frequency detector is highly sensitive to an edge. When an edge of a reference clock cannot be discriminated due to a noise, the phase-frequency detector is hung-up to some state. On the other hand, in a phase detector based on an Exclusive OR circuit, even if an edge cannot be discriminated, the average output is 0 (zero). Therefore,
(iv) The phase-frequency detector is sensitive to a noise.
A charge pump circuit converts logical signals UP and DOWN from the phase-frequency detector (PFD) into specific analog signal levels (ip, xe2x88x92ip and 0). The reason for the conversion is that, since signal amplitude in a digital circuit has a large allowance width, a conversion to a specific analog signal level is necessary. (For example, see: Floyd M. Gardner; xe2x80x9cPhaselock Techniquesxe2x80x9d, 2nd edition, John Wiley and Sons, 1979; and Heinrich Meyr and Gerd Ascheid; xe2x80x9cSynchronization in Digital Communicationsxe2x80x9d, vol. 1, John Wiley and Sons, 1990.)
As shown in FIG. 8A, a charge pump circuit comprises two current sources. In this case, in order to simplify the model circuit, it is assumed that each of the current sources has the same current value Ip. Further, in order to simply describe an output current ip of the charge pump circuit, a negative pulse width is introduced as shown in FIG. 8B. The logical signals UP and DOWN open/close switches S1 and S2, respectively. That is, the logical signal UP closes the switch S1 during a time period of positive pulse width xcfx84 and the logical signal DOWN closes the switch S2 during a time period of negative pulse width xcfx84. Therefore, the output current ip is represented, during the time period of pulse width xcfx84, by the following equation.
ip=Ipsgn(xcfx84)xe2x80x83xe2x80x83(2.1.1)
Otherwise, the output current ip is as follows.
ip=0xe2x80x83xe2x80x83(2.1.2)
(For example, see: Mark Van Paemel; xe2x80x9cAnalysis of Charge-Pump PLL: A New Modelxe2x80x9d, IEEE Trans. Commun., vol. 42, pp. 2490-2498, 1994.)
In this case, sgn(xcfx84) is a sign function. The function sgn(xcfx84) takes a value of +1 when xcfx84 is positive, and takes a value of xe2x88x921 when xcfx84 is negative. When the two switches S1 and S2 are open, no current flows. Therefore, the output node is in high impedance.
A loop filter converts a current ip of the charge pump circuit into an analog voltage value VCTRL. As shown in FIG. 9A, a first order loop filter can be constructed when a resister R2 and a capacitor C are connected in series. When a constant current ip given by the equations (2.1.1) and (2.1.2) is inputted to the filter, an electric charge proportional to a time length is charged in the capacitor C. That is, as shown in FIG. 9B, the control voltage VCTRL linearly changes during the time period xcfx84. In the other time period, the control voltage VCTRL remains constant (for example, see the literature of Mark Van Paemel).                                                                                                               V                    CTRL                                    ⁡                                      (                    t                    )                                                  =                                                                            1                      C                                        ⁢                                                                  ∫                                                  t                          0                                                t                                            ⁢                                                                                                    i                            p                                                    ⁡                                                      (                            τ                            )                                                                          ⁢                                                  xe2x80x83                                                ⁢                                                  ⅆ                          τ                                                                                                      +                                                            V                      CTRL                                        ⁡                                          (                                              t                        0                                            )                                                                                  ,                                                                                                            V                  CTRL                                ⁡                                  (                  t                  )                                            =                                                                    I                    p                                    ⁢                                      R                    2                                                  +                                                                            I                      p                                        C                                    ⁢                                      (                                          t                      -                                              t                        0                                                              )                                                  +                                                      V                    CTRL                                    ⁡                                      (                                          t                      0                                        )                                                                                                          (        2.2        )            
The resistance value and the capacitance value of the loop filter are selected such that an attenuation coefficient and a natural frequency are optimized. (For example, see: Jose Alvarez, Hector Sanchez, Gianfranco Gerosa and Roger Countryman; xe2x80x9cA Wide-bandwidth Low-voltage PLL for Power PC Microprocessorsxe2x80x9d, IEEE J. Solid-State Circuits, vol. 30, pp. 383-391, 1995; and Behzad Razavi; xe2x80x9cMonolithic Phase-Locked Loops and Clock Recovery Circuits: Theory and Designxe2x80x9d, IEEE Press, 1996.) In the present invention, the loop filter is configured as a passive lag filter as shown in FIG. 10 in accordance with a thesis by Ronald E. Best listed below. (See: Ronald E. Best; xe2x80x9cPhase-Locked Loopsxe2x80x9d, 3rd edition, McGraw-Hill, 1997.) Because, as disclosed in this Ronald E. Best""s publication, the combination of a phase-frequency detector and a passive lag filter has infinite pull-in range and hold range, and hence there is no merit in using an other type of filter. In FIG. 10, C=250 pF, R1=920 xcexa9, and R2=360 xcexa9 are set. The VCO is constituted, as shown in FIG. 11, by thirteen stages of CMOS inverters IN-1, IN-2, . . . and IN-13. The power supply voltage is 5 V.
The linear characteristic of the voltage controlled oscillator VCO is given by the following equation.
fVCO=KVCOVCTRLxe2x80x83xe2x80x83(2.3)
In this case, KVCO is a gain of the VCO, and its unit is Hz/V.
When the PLL is in synchronous state (a state that a rise edge of a reference clock accords with a rise edge of a PLL clock), the phase-frequency detector outputs no signal. The charge pump circuit, the loop filter and the VCO provided in the rear stages of the PLL do not send/receive signals and keep maintain the internal state unchanged. On the contrary, when a rise edge of a reference clock does not accord with a rise edge of a PLL clock (in asynchronous state), the phase-frequency detector outputs an UP signal or a DOWN signal to change the oscillation frequency of the VCO. As a result, the charge pump circuit, the loop filter and the VCO provided in the rear stages of the PLL.circuit send/receive signals and transit into a corresponding state. Therefore, it could be understood, in order to measure an internal noise of the PLL circuit, that the PLL circuit must be placed in a synchronous state. On the other hand, in order to test a short-circuit failure or a delay failure of the PLL circuit, the PLL circuit must be moved into other state.
Now, a random jitter will be described.
A jitter on a clock appears as a fluctuation of a rise time and a fall time of a clock pulse series. For this reason, in the transmission of a clock signal, the receiving time or the pulse width of the clock pulse becomes uncertain. (For example, see: Ron K. Poon; xe2x80x9cComputer Circuits Electrical Designxe2x80x9d, Prentice-Hall, Inc, 1995.) FIG. 12 shows jitters of a rise time period and a fall time period of a clock pulse series.
Any component in the blocks shown in FIG. 4 has a potential to cause a jitter. Among those components, the largest factors of a jitter are a thermal noise and a shot noise of the inverters composing the VCO. (For example, see: Todd C. Weigandt, Beomsup Kim and Paul R. Gray; xe2x80x9cAnalysis of Timing Jitter in CMOS Ring Oscillatorsxe2x80x9d, International Symposium on Circuits and System, 1994.) Therefore, the jitter generated from the VCO is a random fluctuation and does not depend on the input. In the present invention, assuming that the major jitter source is the VCO, it is considered that the measurement of a random jitter of an oscillation waveform of the VCO is the most important problem to be solved.
In order to measure only a random jitter of an oscillation waveform of the VCO, it is necessary that the PLL circuit maintains the components other than the VCO to be inactive. Therefore, as mentioned above, it is important that a reference input signal to be supplied to the PLL circuit strictly maintains a constant period so that the PLL circuit under test does not induce a phase error. A concept of this measuring method is shown in FIG. 13.
As a preparation for discussing a phase noise, a zero crossing is defined. Assuming that the minimum value xe2x88x92A of a cosine wave A cos(2xcfx80f0t) is 0% and the maximum value +A thereof is 100%, a level of 50% corresponds to zero amplitude. A point where the waveform crosses a zero level is called a zero crossing.
A phase noise will be discussed with reference to, as an example, a cosine wave generated from an oscillator. An output signal XIDEAL(t) of an ideal oscillator is an ideal cosine wave having no distortion.
XIDEAL(t)=Ac cos(2xcfx80fct+xcex8c)xe2x80x83xe2x80x83(2.4)
In this case, AC and fC are nominal values of amplitude and a frequency, respectively, and xcex8c is an initial phase angle. When the output signal XIDEAL(t) is observed in frequency domain, the output signal is measured as a line spectrum as shown in FIG. 14. In the actual oscillator, there are some differences from the nominal values. In this case, the output signal is expressed as follows.
xe2x80x83XOSC(t)=[Ac+xcex5(t)]cos(2xcfx80fct+xcex8c+xcex94xcfx86(t))xe2x80x83xe2x80x83(2.5.1)
XOSC(t)=Ac cos(2xcfx80fct+xcex8c+xcex94xcfx86(t))xe2x80x83xe2x80x83(2.5.2)
In the above equations, xcex5(t) represents a fluctuation of amplitude. In the present invention, the discussion will be made assuming that, as shown in the equation (2.5.2), the amplitude fluctuation xcex5(t) of the oscillator is zero. In the above equations, xcex94xcfx86(t) represents a phase fluctuation. That is, xcex94xcfx86(t) is a term for modulating the ideal cosine wave. The initial phase angle xcex8C follows a uniform distribution in the range of an interval (0,2xcfx80). On the other hand, the phase fluctuation xcex94xcfx86(t) is a random data and follows, for example, a Gaussian distribution. This xcex94xcfx86(t) is called a phase noise.
In FIG. 15, an output signal XIDEAL(t) of an ideal oscillator and an output signal XOSC(t) of an actual oscillator are plotted. Comparing those signals with one another, it can be seen that the zero crossing of XOSC(t) is changed due to xcex94xcfx86(t).
On the other hand, as shown in FIG. 16, when the oscillation signal XOSC(t) is transformed into frequency domain, the influence of a phase noise is observed as a spectrum diffusion in the proximity of the nominal frequency f0. Comparing FIG. 15 with FIG. 16, it can be said that frequency domain is easier to observe the influence of a phase noise. However, even if the clock pulse shown in FIG. 12 is transformed into frequency domain, the maximum value of the pulse width fluctuation cannot be estimated. Because, the transformation is a process for averaging the fluctuation in certain frequencies, and in the summing step of the process, the maximum value and the minimum value are mutually canceled. Therefore, in a peak-to-peak jitter estimating method which is an object of the present invention, a process in time domain must be a nucleus of the method.
Here, it will be made clear that an additive noise at the reference input end to the PLL circuit is equal to an additive noise at the input end of the loop filter. (See: Floyd M. Gardner; xe2x80x9cPhaselock Techniquesxe2x80x9d, 2nd edition, John Wiley and Sons, 1979; and John G. Proakis; xe2x80x9cDigital Communicationsxe2x80x9d, 2nd edition, McGraw-Hill, 1989.) FIG. 17 shows an additive noise at the reference input end to the PLL circuit. In order to simplify the calculation, it is assumed that a phase detector of the PLL circuit is a sine wave phase detector (mixer).
The PLL circuit is phase-synchronized with a given reference signal expressed by the following equation (2.6).
Xref(t)=Ac cos(2xcfx80fct)xe2x80x83xe2x80x83(2.6)
In this case, it is assumed that the following additive noise expressed by the equation (2.7) is added to this reference signal Xref(t).
Xnoise(t)=ni(t)cos(2xcfx80fct)xe2x88x92nq(t)sin(2xcfx80fct)xe2x80x83xe2x80x83(2.7)
XVCO(t)=cos(2xcfx80fct+xcex94xcfx86)xe2x80x83xe2x80x83(2.8)
An oscillation waveform of the VCO expressed by the above equation (2.8) and the reference signal Xref(t)+Xnoise(t) are inputted to the phase detector to be converted to a difference frequency component.                                                                                           x                  PD                                ⁡                                  (                  t                  )                                            =                                                K                  PD                                ⁡                                  (                                                                                                              A                          c                                                2                                            ⁢                                              cos                        ⁡                                                  (                          Δφ                          )                                                                                      +                                                                                                                        n                            i                                                    ⁡                                                      (                            t                            )                                                                          2                                            ⁢                                              cos                        ⁡                                                  (                          Δφ                          )                                                                                      -                                                                                                                        n                            q                                                    ⁡                                                      (                            t                            )                                                                          2                                            ⁢                                              sin                        ⁡                                                  (                          Δφ                          )                                                                                                      )                                                                                                        =                                                                    K                    PD                                    2                                ⁢                                                      A                    c                                    [                                                            cos                      ⁡                                              (                        Δφ                        )                                                              +                                          (                                                                                                                                                                  n                                i                                                            ⁡                                                              (                                t                                )                                                                                                                    A                              c                                                                                ⁢                                                      cos                            ⁡                                                          (                              Δφ                              )                                                                                                      -                                                                                                                                            n                                q                                                            ⁡                                                              (                                t                                )                                                                                                                    A                              c                                                                                ⁢                                                      sin                            ⁡                                                          (                              Δφ                              )                                                                                                                          ]                                                                                                                              (        2.9        )            
In this case, KPD is a gain of a phase comparator. Therefore, it can be understood that the additive noise of the reference signal is equal to that an additive noise expressed by the following equation (2.10) is applied to an input end of the loop filter.                                           x                          noise              ,              LPF                                ⁡                      (            t            )                          =                                                                              n                  i                                ⁡                                  (                  t                  )                                                            A                c                                      ⁢                          cos              ⁡                              (                Δφ                )                                              -                                                                      n                  q                                ⁡                                  (                  t                  )                                                            A                c                                      ⁢                          sin              ⁡                              (                Δφ                )                                                                        (        2.10        )            
FIG. 18 shows an additive noise at the input end of the loop filter. If a power spectrum density of the additive noise at the reference input end of the PLL circuit is assumed to be N0[V2/Hz], the power spectrum density Gnn(f) of the additive noise at the input end of this loop filter is, from the equation (2.10), expressed by the following equation (2.11).                                           G            nn                    ⁡                      (            f            )                          =                                            2              ⁢                              N                0                                                    A              c              2                                ⁡                      [                                          V                2                            /              Hz                        ]                                              (        2.11        )            
Moreover, it can be seen from the equation (2.9) that when a phase difference xcex94xcfx86 between the oscillation waveform of the VCO and the reference signal becomes xcfx80/2, an output of the phase detector becomes zero. That is, if a sine wave phase detector is used, when the phase of the VCO is shifted by 90 degrees from the phase of the reference signal, the VCO is phase-synchronized with the reference signal. Further, in this calculation, the additive noise is neglected.
Next, using a model of equivalent additive noise shown in FIG. 17, an amount of jitter produced by an additive noise will be made clear. (See: Heinrich Meyr and Gerd Ascheid; xe2x80x9cSynchronization in Digital Communicationsxe2x80x9d, vol. 1, John Wiley and Sons, 1990.) In order to simplify the expression, assuming xcex8i=0, the phase xcex80 of the output signal corresponds to an error. A phase spectrum of the oscillation waveform of the VCO is expressed by the equation (2.12).
Gxcex8Oxcex8O(f)=|H(f)|2Gnn(f)xe2x80x83xe2x80x83(2.12)
In this case, H(f) is a transfer function of the PLL circuit.                               H          ⁡                      (            s            )                          =                                                            θ                o                            ⁡                              (                s                )                                                                    θ                i                            ⁡                              (                s                )                                              =                                                    K                VCO                            ⁢                              K                PD                            ⁢                              F                ⁡                                  (                  s                  )                                                                                    K                VCO                            ⁢                              K                PD                            ⁢                              F                ⁡                                  (                  s                  )                                                                                        (        2.13        )            
Since a phase error is xe2x88x92xcex80, a variance of the phase error is given by the following equation (2.14).                               σ          Δφ          2                =                              1            π                    ⁢                                    ∫              0              ∞                        ⁢                                                            "LeftBracketingBar"                                      H                    ⁡                                          (                      f                      )                                                        "RightBracketingBar"                                2                            ⁢                                                G                  nn                                ⁡                                  (                  f                  )                                            ⁢                              xe2x80x83                            ⁢                              ⅆ                f                                                                        (        2.14        )            
Substituting the equation (2.11) for the equation (2.14), the following two equations are obtained.                               σ          Δφ          2                =                                            2              ⁢                              N                0                                                    A              c              2                                ⁢                      B            e                                              (                  2.15          ⁢          .1                )                                          σ          Δφ          2                =                  1                                                    (                                                      A                    c                                                        2                                                  )                            2                                                      N                0                            ⁢                              B                e                                                                        (                  2.15          ⁢          .2                )            
That is, if a signal to noise ratio of the loop             (                        A          c                          2                    )        2              N      0        ⁢          B      e      
is large, a phase noise becomes small. In this case, Be is an equivalent noise band width of the loop.
As described above, an additive noise at the reference input end of the PLL circuit or an additive noise at the input end of the loop filter is observed as an output phase noise, which is a component passed through a lowpass filter corresponding to the loop characteristic. The power of a phase noise is inversely proportional to a signal to noise ratio of the PLL loop.
Next, a discussion will be made as to how a phase fluctuation due to an internal noise of the VCO influences a phase of output signal of the PLL. (See: Heinrich Meyr and Gerd Ascheid; xe2x80x9cSynchronization in Digital Communicationsxe2x80x9d, vol. 1, John Wiley and Sons, 1990.) An output signal of the VCO is assumed to be expressed by the following equation (2.16).
XVCO, noise=Ac cos(2xcfx80fct+xcex8p(t)+xcexa8(t))xe2x80x83xe2x80x83(2.16)
In this case, xcex8p(t) is a phase of an ideal VCO. An internal thermal noise or the like generates xcexa8(t). The generated xcexa8(t) is an internal phase noise and randomly fluctuates the phase of the VCO. FIG. 19 shows an internal phase noise model of the VCO. A phase xcex8p(S) at the output end of the ideal VCO is given by a equation (2.17).                                           θ            p                    ⁡                      (            s            )                          =                              K            PD                    ⁢                      K            VCO                    ⁢                                    F              ⁡                              (                s                )                                      s                    ⁢                      Φ            ⁡                          (              s              )                                                          (        2.17        )            
In this case, "PHgr"(t) is a phase error and corresponds to an output of the phase detector.
"PHgr"(s)=xcex8i(s)xe2x88x92xcex8o(s)=xcex8i(s)xe2x88x92(xcex8p(s)+xcexa8(s))xe2x80x83xe2x80x83(2.18)
Substituting xcex8p(S) of the equation (2.17) for that of the equation (2.18), the following equation (2.19) is obtained.                               Φ          ⁡                      (            s            )                          =                                            θ              i                        ⁡                          (              s              )                                -                      [                                                                                                      K                      PD                                        ⁢                                          K                      VCO                                        ⁢                                          F                      ⁡                                              (                        s                        )                                                                              s                                ⁢                                  Φ                  ⁡                                      (                    s                    )                                                              +                              Ψ                ⁡                                  (                  s                  )                                                      ]                                              (        2.19        )            
The following equation (2.20.1) can be obtained by rearranging the above equation (2.19).                               Φ          ⁡                      (            s            )                          =                              1                          1              +                                                                    K                    PD                                    ⁢                                      K                    VCO                                    ⁢                                      F                    ⁡                                          (                      s                      )                                                                      s                                              ⁢                      (                                                            θ                  i                                ⁡                                  (                  s                  )                                            -                              Ψ                ⁡                                  (                  s                  )                                                      )                                              (                  2.20          ⁢          .1                )            
Substituting the equation (2.13) for the equation (2.20.1), the following equation (2.20.2) is obtained.
"PHgr"(s)=(1xe2x88x92H(s))(xcex8i(s)xe2x88x92xcexa8(s))xe2x80x83xe2x80x83(2.20.2)
Therefore, a phase fluctuation due to an internal noise of the VCO is expressed by the following equation (2.21).                               σ          Φ          2                =                              1            π                    ⁢                                    ∫              0              ∞                        ⁢                                                            "LeftBracketingBar"                                      1                    -                                          H                      ⁡                                              (                        f                        )                                                                              "RightBracketingBar"                                2                            ⁢                                                G                  ΨΨ                                ⁡                                  (                  f                  )                                            ⁢                              xe2x80x83                            ⁢                              ⅆ                f                                                                        (        2.21        )            
That is, an internal phase noise of the VCO is observed as a phase noise of an output signal of the PLL circuit, which is a component passed through a highpass filter. This highpass filter corresponds to a phase error transfer function of the loop.
As stated above, an internal thermal noise of the VCO becomes a phase noise of an oscillation waveform of the VCO. Further, a component passed through the highpass filter corresponding to a loop phase error is observed as an output phase noise.
An additive noise of the PLL circuit and/or an internal thermal noise of the VCO is converted to a phase noise of an oscillation waveform of the VCO. An additive noise of the PLL circuit and/or an internal thermal noise of the VCO is observed, correspondingly to the path from a block generating a noise through the output of the PLL circuit, as a phase noise of a low frequency component or a high frequency component. Therefore, it can be seen that a noise of the PLL circuit has an effect to give a fluctuation to a phase of an oscillation waveform of the VCO. This is equivalent to a voltage change at the input end of the VCO. In the present invention, an additive noise is applied to the input end of the VCO to randomly modulate the phase of a waveform of the VCO so that a jitter is simulated. FIG. 20 shows a method of simulating a jitter.
Next, a method of measuring a jitter of a clock will be explained. A peak-to-peak jitter is measured in time domain and an RMS jitter is measured in frequency domain. Each of those conventional jitter measuring methods requires approximately 10 minutes of test time. On the other hand, in the case of a VLSI test, only approximately 100 msec of test time is allocated to one test item. Therefore, the conventional method of measuring a jitter cannot be applied to a test in the VLSI production line.
In the study of the method of measuring a jitter, the zero crossing is an important concept. From the view point of period measurement, a relationship between the zero crossings of a waveform and the zero crossings of the fundamental waveform of its fundamental frequency will be discussed. It will be proven that xe2x80x9cthe waveform of its fundamental frequency contains the zero-crossing information of the original waveformxe2x80x9d. In the present invention, this characteristic of the fundamental waveform is referred to as xe2x80x9ctheorem of zero crossing.xe2x80x9d. An explanation will be given on an ideal clock waveform Xd50%(t) shown in FIG. 21, as an example, having 50% duty cycle. Assuming that a period of this clock waveform is T0, the Fourier transform of the clock waveform is given by the following equation (3.1). (For example, refer to a reference literature c1.)                                           S                          d50              ⁢              %                                ⁡                      (            f            )                          =                              ∑                          k              =                              -                ∞                                                    +              ∞                                ⁢                      xe2x80x83                    ⁢                                                    2                ⁢                                  sin                  ⁡                                      (                                                                  π                        ⁢                                                  xe2x80x83                                                ⁢                        k                                            2                                        )                                                              k                        ⁢                          δ              ⁡                              (                                  f                  -                                      kf                    0                                                  )                                                                        (        3.1        )            
That is, a period of the fundamental wave is equal to a period of the clock.                               T          0                =                              1            f                    ⁢                      δ            ⁡                          (                              f                -                                  f                  0                                            )                                                          (        3.2        )            
When the fundamental waveform of the clock signal is extracted, its zero crossings correspond to the zero crossings of the original clock waveform. Therefore, a period of a clock waveform can be estimated from the zero crossings of its fundamental waveform. In this case, the estimation accuracy is not improved even if some harmonics are added to the fundamental waveform. However, harmonics and an estimation accuracy of a period will be verified later.
Next, Hilbert transform and an analytic signal will be explained (for example, refer to a reference literature c2).
As can be seen from the equation (3.1), when the Fourier transform of the waveform Xa(t) is calculated, a power spectrum Saa(f) ranging from negative frequencies through positive frequencies can be obtained. This is called a two-sided power spectrum. The negative frequency spectrum is a mirror image of the positive frequency spectrum about an axis of f=0. Therefore, the two-sided power spectrum is symmetric about the axis of f=0, i.e., Saa(xe2x88x92f)=Saa(f). However, the spectrum of negative frequencies cannot be observed. There can be defined a spectrum Gaa(f) in which negative frequencies are cut to zero and, instead, observable positive frequencies are doubled. This is called one-sided power spectrum.
Gaa(f)=2Saa(f)f greater than 0
Gaa(f)=0 f less than 0xe2x80x83xe2x80x83(3.3.1)
Gaa(f)=Saa(f)[1+sgn(f)]xe2x80x83xe2x80x83(3.3.2)
In this case, sgn(f) is a sign function, which takes a value of +1 when f is positive and takes a value of xe2x88x921 when f is negative. This one-sided spectrum corresponds to a spectrum of an analytic signal z(t). The analytic signal z(t) can be expressed in time domain as follows.                               z          ⁡                      (            t            )                          =                                            x              a                        ⁡                          (              t              )                                +                      j            ⁢                                                            x                  ⋒                                a                            ⁡                              (                t                )                                                                        (        3.4        )                                                                    x              ⋒                        a                    ⁡                      (            t            )                          =                              H            ⁡                          [                                                x                  a                                ⁡                                  (                  t                  )                                            ]                                =                                    1              π                        ⁢                                          ∫                                  -                  ∞                                                  +                  ∞                                            ⁢                                                                                          x                      a                                        ⁡                                          (                      τ                      )                                                                            t                    -                    τ                                                  ⁢                                  xe2x80x83                                ⁢                                  ⅆ                  τ                                                                                        (        3.5        )            
The real part corresponds to the original waveform Xa(t). The imaginary part is given by the Hilbert transform {circumflex over (x)}a(t) of the original waveform. As shown by the equation (3.5), the Hilbert transform             x      ⋒        a    ⁡      (    t    )  
of a waveform Xa(t) is given by a convolution of the waveform Xa(t) and   1      π    ⁢          xe2x80x83        ⁢    t  
Let""s obtain the Hilbert transform of a waveform handled in the present invention. First, the Hilbert transform of a cosine wave is derived.                               H          ⁡                      [                          cos              ⁢                              xe2x80x83                            ⁢                              (                                  2                  ⁢                                      xe2x80x83                                    ⁢                  π                  ⁢                                      xe2x80x83                                    ⁢                                      f                    0                                    ⁢                                      xe2x80x83                                    ⁢                  t                                )                                      ]                          =                ⁢                              -                          1              π                                ⁢                      xe2x80x83                    ⁢                                    ∫                              -                ∞                                            +                ∞                                      ⁢                                                            cos                  ⁢                                      xe2x80x83                                    ⁢                                      (                                          2                      ⁢                                              xe2x80x83                                            ⁢                      π                      ⁢                                              xe2x80x83                                            ⁢                                              f                        0                                            ⁢                                              xe2x80x83                                            ⁢                      τ                                        )                                                                    τ                  -                  t                                            ⁢                              xe2x80x83                            ⁢                              ⅆ                τ                                                                            =                ⁢                              -                          1              π                                ⁢                      xe2x80x83                    ⁢                                    ∫                              -                ∞                                            +                ∞                                      ⁢                                                            cos                  ⁢                                      xe2x80x83                                    ⁢                                      (                                          2                      ⁢                                              xe2x80x83                                            ⁢                      π                      ⁢                                              xe2x80x83                                            ⁢                                              f                        0                                            ⁢                                              xe2x80x83                                            ⁢                                              (                                                  y                          +                          t                                                )                                                              )                                                  y                            ⁢                              xe2x80x83                            ⁢                              ⅆ                y                                                                            =                ⁢                              1            π                    [                                    cos              ⁢                              xe2x80x83                            ⁢                              (                                  2                  ⁢                                      xe2x80x83                                    ⁢                  π                  ⁢                                      xe2x80x83                                    ⁢                                      f                    0                                    ⁢                                      xe2x80x83                                    ⁢                  t                                )                            ⁢                              xe2x80x83                            ⁢                                                ∫                                      -                    ∞                                                        +                    ∞                                                  ⁢                                                                            cos                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                  2                          ⁢                                                      xe2x80x83                                                    ⁢                          π                          ⁢                                                      xe2x80x83                                                    ⁢                                                      f                            0                                                    ⁢                                                      xe2x80x83                                                    ⁢                          y                                                )                                                              y                                    ⁢                                      xe2x80x83                                    ⁢                                      ⅆ                    y                                                                        -                                                          ⁢                  sin          ⁢                      xe2x80x83                    ⁢                      (                          2              ⁢                              xe2x80x83                            ⁢              π              ⁢                              xe2x80x83                            ⁢                              f                0                            ⁢                              xe2x80x83                            ⁢              t                        )                    ⁢                      xe2x80x83                    ⁢                                    ∫                              -                ∞                                            +                ∞                                      ⁢                                                            cos                  ⁢                                      xe2x80x83                                    ⁢                                      (                                          2                      ⁢                                              xe2x80x83                                            ⁢                      π                      ⁢                                              xe2x80x83                                            ⁢                                              f                        0                                            ⁢                                              xe2x80x83                                            ⁢                      y                                        )                                                  y                            ⁢                              xe2x80x83                            ⁢                              ⅆ                y                                                    ]            
Since the integral of the first term is equal to zero and the integral of the second term is xcfx80, the,following equation (3.6) is obtained.
H[cos(2xcfx80f0t)]=sin(2xcfx80f0t)xe2x80x83xe2x80x83(3.6)
Similarly, the following equation (3.7) is obtained.
H[sin(2xcfx80f0t)]=xe2x88x92cos(2xcfx80f0t)xe2x80x83xe2x80x83(3.7)
Next, the Hilbert transform of a square wave corresponding to a clock waveform will be derived (for example, refer to a reference literature c3). The Fourier series of an ideal clock waveform shown in FIG. 21 is given by the following equation (3.8).                                           x                          d50              ⁢              %                                ⁢                      xe2x80x83                    ⁢                      (            t            )                          =                              1            2                    +                                    2              π                        ⁡                          [                                                cos                  ⁢                                      xe2x80x83                                    ⁢                                                            2                      ⁢                                              xe2x80x83                                            ⁢                      π                                                              T                      0                                                        ⁢                                      xe2x80x83                                    ⁢                  t                                -                                                      1                    3                                    ⁢                                      xe2x80x83                                    ⁢                  cos                  ⁢                                      xe2x80x83                                    ⁢                  3                  ⁢                                      xe2x80x83                                    ⁢                                                            2                      ⁢                                              xe2x80x83                                            ⁢                      π                                                              T                      0                                                        ⁢                                      xe2x80x83                                    ⁢                  t                                +                                                      1                    5                                    ⁢                                      xe2x80x83                                    ⁢                  cos                  ⁢                                      xe2x80x83                                    ⁢                  5                  ⁢                                      xe2x80x83                                    ⁢                                                            2                      ⁢                                              xe2x80x83                                            ⁢                      π                                                              T                      0                                                        ⁢                                      xe2x80x83                                    ⁢                  t                                -                …                            ]                                                          (        3.8        )            
The Hilbert transform is given, using the equation (3.6), by the following equation (3.9).                               H          ⁡                      [                                          x                                  d50                  ⁢                                      xe2x80x83                                    ⁢                  %                                            ⁢                              xe2x80x83                            ⁢                              (                t                )                                      ]                          =                              2            π                    ⁡                      [                                          sin                ⁢                                  xe2x80x83                                ⁢                                                      2                    ⁢                                          xe2x80x83                                        ⁢                    π                                                        T                    0                                                  ⁢                                  xe2x80x83                                ⁢                t                            -                                                1                  3                                ⁢                                  xe2x80x83                                ⁢                sin                ⁢                                  xe2x80x83                                ⁢                3                ⁢                                  xe2x80x83                                ⁢                                                      2                    ⁢                                          xe2x80x83                                        ⁢                    π                                                        T                    0                                                  ⁢                                  xe2x80x83                                ⁢                t                            +                                                1                  5                                ⁢                                  xe2x80x83                                ⁢                sin                ⁢                                  xe2x80x83                                ⁢                5                ⁢                                  xe2x80x83                                ⁢                                                      2                    ⁢                                          xe2x80x83                                        ⁢                    π                                                        T                    0                                                  ⁢                                  xe2x80x83                                ⁢                t                            -              …                        ]                                              (        3.9        )            
FIG. 22 shows examples of a clock waveform and its Hilbert transform. Those waveforms are based on the partial summation up to the 11th-order harmonics, respectively. The period T0 in this example is 20 nsec.
An analytic signal z(t) is introduced by J. Dugundji to uniquely obtain an envelope of a waveform. (For example, refer to a reference literature c4.) If an analytic signal is expressed in a polar coordinate system, the following equations (3.10.1), (3.10.2) and (3.10.3) are obtained.                               z          ⁢                      xe2x80x83                    ⁢                      (            t            )                          =                  A          ⁢                      xe2x80x83                    ⁢                      (            t            )                    ⁢                      xe2x80x83                    ⁢                      ⅇ                          j              ⁢                              xe2x80x83                            ⁢              Θ              ⁢                              xe2x80x83                            ⁢                              (                t                )                                                                        (                  3.10          ⁢          .1                )                                          A          ⁢                      xe2x80x83                    ⁢                      (            t            )                          =                                                            x                a                2                            ⁢                              xe2x80x83                            ⁢                              (                t                )                                      +                                                            x                  ^                                a                2                            ⁢                              xe2x80x83                            ⁢                              (                t                )                                                                        (                  3.10          ⁢          .2                )                                          Θ          ⁢                      xe2x80x83                    ⁢                      (            t            )                          =                              tan                          -              1                                ⁡                      [                                                                                x                    ⋒                                    a                                ⁢                                  xe2x80x83                                ⁢                                  (                  t                  )                                                                              x                  a                                ⁢                                  xe2x80x83                                ⁢                                  (                  t                  )                                                      ]                                              (                  3.10          ⁢          .3                )            
In this case, A(t) represents an envelope of Xa(t). For this reason, z(t) is called pre-envelope by J. Dugundji. Further, "THgr"(t) represents an instantaneous phase of Xa(t). In the method of measuring a jitter according to the present invention, a method of estimating this instantaneous phase is the nucleus.
If a measured waveform is handled as a complex number, its envelope and instantaneous phase can simply be obtained. Hilbert transform is a tool for transforming a waveform to an analytic signal. An analytic signal can be obtained by the procedure of the following Algorithm 1.
Algorithm 1 (Procedure for transforming a real waveform to an analytic signal):
1. A waveform is transformed to a frequency domain using fast Fourier transform;
2. Negative frequency components are cut to zero and positive frequency components are doubled; and
3. The spectrum is transformed to a time domain using inverse fast Fourier transform.
Next, a phase unwrap method for converting a phase to a continuous phase will briefly be described.
The phase unwrap method is a method proposed to obtain a complex cepstrum. (For example, refer to a reference literature c5.) When a complex logarithmic function log(z) is defined as an arbitrary complex number satisfying elog(z)=z, the following equation (3.11) can be obtained. (For example, refer to a reference literature c6.)
log(z)=log|z|+jARG(z)xe2x80x83xe2x80x83(3.11)
The result of Fourier transform of a time waveform Xa(n) is assumed to be Sa(ejxcfx89). When its logarithmic magnitude spectrum log|Sa(ejxcfx89)| and phase spectrum ARG[Sa(ejxcfx89)] correspond to a real part and an imaginary part of a complex spectrum, respectively, and inverse Fourier transform is applied, a complex cepstrum Ca(n) can be obtained.                                                                                           c                  a                                ⁢                                  xe2x80x83                                ⁢                                  (                  n                  )                                            =                            ⁢                                                1                                      2                    ⁢                                          xe2x80x83                                        ⁢                    π                                                  ⁢                                  xe2x80x83                                ⁢                                                      ∫                                          -                      π                                                              +                      π                                                        ⁢                                                            log                      ⁡                                              [                                                                              S                            a                                                    ⁢                                                      xe2x80x83                                                    ⁢                                                      (                                                          ⅇ                                                              j                                ⁢                                                                  xe2x80x83                                                                ⁢                                ω                                                                                      )                                                                          ]                                                              ⁢                                          ⅇ                                              j                        ⁢                                                  xe2x80x83                                                ⁢                        ω                        ⁢                                                  xe2x80x83                                                ⁢                        n                                                              ⁢                                          xe2x80x83                                        ⁢                                          ⅆ                      ω                                                                                                                                              =                            ⁢                                                1                                      2                    ⁢                                          xe2x80x83                                        ⁢                    π                                                  ⁢                                  xe2x80x83                                ⁢                                                      ∫                                          -                      π                                                              +                      π                                                        ⁢                                                            {                                                                        log                          ⁢                                                      "LeftBracketingBar"                                                                                          S                                a                                                            ⁢                                                              xe2x80x83                                                            ⁢                                                              (                                                                  ⅇ                                                                      j                                    ⁢                                                                          xe2x80x83                                                                        ⁢                                    ω                                                                                                  )                                                                                      "RightBracketingBar"                                                                          +                                                                              j                            ⁢                            ARG                                                    ⁡                                                      [                                                                                          S                                a                                                            ⁢                                                              xe2x80x83                                                            ⁢                                                              (                                                                  ⅇ                                                                      j                                    ⁢                                                                          xe2x80x83                                                                        ⁢                                    ω                                                                                                  )                                                                                      ]                                                                                              }                                        ⁢                                          ⅇ                                              j                        ⁢                                                  xe2x80x83                                                ⁢                        ω                        ⁢                                                  xe2x80x83                                                ⁢                        n                                                              ⁢                                          xe2x80x83                                        ⁢                                          ⅆ                      ω                                                                                                                              (        3.12        )            
In this case, ARG represents the principal value of the phase. The principal value of the phase lies in the range [xe2x88x92xcfx80, xcfx80]. There exist discontinuity points at xe2x88x92xcfx80 and +xcfx80 in the phase spectrum of the 2nd term. Since an influence of those discontinuity points diffuses throughout entire time domain by the application of inverse Fourier transform, a complex cepstrum cannot accurately be estimated. In order to convert a phase to a continuous phase, an unwrapped phase is introduced. An unwrapped phase can be uniquely given by integrating a derived function of a phase.                               arg          ⁡                      [                                          S                a                            ⁢                              xe2x80x83                            ⁢                              (                                  ⅇ                                      j                    ⁢                                          xe2x80x83                                        ⁢                    ω                                                  )                                      ]                          =                              ∫            0            ω                    ⁢                                                                      ⅆ                  ARG                                ⁢                                  ⌊                                                            S                      a                                        ⁢                                          xe2x80x83                                        ⁢                                          (                                              ⅇ                                                  j                          ⁢                                                      xe2x80x83                                                    ⁢                          η                                                                    )                                                        ⌋                                                            ⅆ                η                                      ⁢                          xe2x80x83                        ⁢                          ⅆ              η                                                          (                  3.13          ⁢          .1                )                                          arg          ⁢                      xe2x80x83                    ⁢                      ⌊                                          S                a                            ⁢                              xe2x80x83                            ⁢                              (                                  ⅇ                                      j                    ⁢                                          xe2x80x83                                        ⁢                    0                                                  )                                      ⌋                          =        0                            (                  3.13          ⁢          .2                )            
Where, arg represents an unwrapped phase. An algorithm for obtaining an unwrapped phase by removing discontinuity points from a phase spectrum in frequency domain has been developed by Ronald W. Schafer and Donald G. Childers (for example, refer to a reference literature c7).
Algorithm 2:                     1        ⁢                  xe2x80x83                ⁢        ARG        ⁢                  xe2x80x83                ⁢                  (          0          )                    =      0        ,                  C        ⁢                  xe2x80x83                ⁢                  (          0          )                    =      0                  2      ⁢              xe2x80x83            ⁢      C      ⁢              xe2x80x83            ⁢              (        k        )              =          {                                                                                                                              C                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                  k                          -                          1                                                )                                                              -                                          2                      ⁢                                              xe2x80x83                                            ⁢                      π                                                        ,                                                                                                                        if                      ⁢                                              xe2x80x83                                            ⁢                      ARG                      ⁢                                              xe2x80x83                                            ⁢                                              (                        k                        )                                                              -                                          ARG                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                  k                          -                          1                                                )                                                                               greater than                   π                                                                                                                                                C                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                  k                          -                          1                                                )                                                              +                                          2                      ⁢                                              xe2x80x83                                            ⁢                      π                                                        ,                                                                                                                        if                      ⁢                                              xe2x80x83                                            ⁢                      ARG                      ⁢                                              xe2x80x83                                            ⁢                                              (                        k                        )                                                              -                                          ARG                      ⁢                                              xe2x80x83                                            ⁢                                              (                                                  k                          -                          1                                                )                                                                               less than                   π                                                                                                                          C                    ⁢                                          xe2x80x83                                        ⁢                                          (                                              k                        -                        1                                            )                                                        ,                                                                              otherwise                  .                                                              ⁢                      
                    ⁢          3          ⁢                      xe2x80x83                    ⁢          arg          ⁢                      xe2x80x83                    ⁢                      (            k            )                          =                              ARG            ⁢                          xe2x80x83                        ⁢                          (              k              )                                +                      C            ⁢                          xe2x80x83                        ⁢                          (              k              )                                          
An unwrapped phase will be obtained by the above Algorithm 2. First, a judgment is made, by obtaining differences between main values of adjacent phases, to see if there is. a discontinuity point. If there is a discontinuity point, xc2x12xcfx80 is added to the main value to remove the discontinuity point from the phase spectrum (refer to the reference literature c7).
In the above algorithm 2, it is assumed that a difference between adjacent phases is smaller than xcfx80. That is, a resolution for observing a phase spectrum is required to be sufficiently small. However, at a frequency in the proximity of a pole (a resonance frequency), the phase difference between the adjacent phases is larger than xcfx80. If a frequency resolution for observing a phase spectrum is rough, it cannot be determined whether or not a phase is increased or decreased by equal to or more than 2xcfx80. As a result, an unwrapped phase cannot be accurately obtained. This problem has been solved by Jose M. Tribolet. That is, Jose M. Tribolet proposed a method wherein the integration of the derived function of a phase in the equation (3.12) is approximated by a numerical integration based on a trapezoidal rule and a division width of the integrating section is adaptively subdivided to fine pieces until an estimated phase value for determining whether or not a phase is increased or decreased by equal to or more than 2xcfx80 is obtained (for example, refer to a reference literature c8). In such a way, an integer 1 of the following equation, (3.14) is found.
arg[Sa(ejxcexa9)]=ARG[Sa(ejxcexa9)]+2xcfx80l(xcexa9)xe2x80x83xe2x80x83(3.14)
The Tribolet""s algorithm has been expanded by Kuno P. Zimmermann to a phase unwrap algorithm in time domain (for example, refer to a reference literature c9).
In the present invention, the phase unwrap is used to convert an instantaneous phase waveform in time domain into a continuous phase by removing discontinuity points at xe2x88x92xcfx80 and +xcfx80 from the instantaneous phase waveform. A sampling condition for uniquely performing the phase unwrap in time domain will be discussed later.
Next, a linear trend estimating method to be utilized to obtain a linear phase from a continuous phase will briefly be described (for example, refer to reference literatures c10 and c11).
The target of the linear trend estimating method is to find a linear phase g(x) adaptable to a phase data yi.
g(x)=a+bxxe2x80x83xe2x80x83(3.15)
In this case, xe2x80x9caxe2x80x9d and xe2x80x9cbxe2x80x9d are the constants to be found. A square error R between g(xi) and each data (xi, yi) is given by the following equation (3.16).                     R        =                              ∑                          i              =              1                        L                    ⁢                      xe2x80x83                    ⁢                                    (                                                y                  i                                -                a                -                                  bx                  i                                            )                        2                                              (        3.16        )            
In this case, L is the number of phase data. A linear phase for minimizing the square error is found. A partial differentiation of the equation (3.16) with respect to each of the unknown constants a and b is calculated and the result is put into zero. Then the following equations (3.17.1) and (3.17.2) can be obtained.                                           ∂            R                                ∂            a                          =                                            ∑                              i                =                1                            L                        ⁢                          xe2x80x83                        ⁢                          (                                                y                  i                                -                a                -                                  bx                  i                                            )                                =          0                                    (                  3.17          ⁢          .1                )                                                      ∂            R                                ∂            b                          =                                            ∑                              i                =                1                            L                        ⁢                          xe2x80x83                        ⁢                                          x                i                            ⁢                              xe2x80x83                            ⁢                              (                                                      y                    i                                    -                  a                  -                                      bx                    i                                                  )                                              =          0                                    (                  3.17          ⁢          .2                )            
Those equations are transformed to obtain the following equation (3.18).                                           [                                                            L                                                                                            ∑                                              xe2x80x83                                                              ⁢                                          xe2x80x83                                        ⁢                                          x                      i                                                                                                                                                              ∑                                              xe2x80x83                                                              ⁢                                          xe2x80x83                                        ⁢                                          x                      i                                                                                                                                  ∑                                              xe2x80x83                                                              ⁢                                          xe2x80x83                                        ⁢                                          x                      i                      2                                                                                            ]                    ⁡                      [                                                            a                                                                              b                                                      ]                          =                  [                                                                                          ∑                                          xe2x80x83                                                        ⁢                                      xe2x80x83                                    ⁢                                      y                    i                                                                                                                                            ∑                                          xe2x80x83                                                        ⁢                                      xe2x80x83                                    ⁢                                                            x                      i                                        ⁢                                          xe2x80x83                                        ⁢                                          y                      i                                                                                                    ]                                    (        3.18        )            
Therefore, the following equation (3.19) can be obtained.                               [                                                    a                                                                    b                                              ]                =                                            1                                                L                  ⁢                                      xe2x80x83                                    ⁢                                                            ∑                                              xe2x80x83                                                              ⁢                                          xe2x80x83                                        ⁢                                          x                      i                      2                                                                      -                                                      (                                                                  ∑                                                  xe2x80x83                                                                    ⁢                                              xe2x80x83                                            ⁢                                              x                        i                        2                                                              )                                    2                                                      ⁡                          [                                                                                                                  ∑                                                  xe2x80x83                                                                    ⁢                                              xe2x80x83                                            ⁢                                              x                        i                        2                                                                                                                        -                                                                        ∑                                                      xe2x80x83                                                                          ⁢                                                  xe2x80x83                                                ⁢                                                  x                          i                                                                                                                                                                                -                                                                        ∑                                                      xe2x80x83                                                                          ⁢                                                  xe2x80x83                                                ⁢                                                  x                          i                                                                                                                          L                                                              ]                                ⁡                      [                                                                                                      ∑                                              xe2x80x83                                                              ⁢                                          xe2x80x83                                        ⁢                                          y                      i                                                                                                                                                              ∑                                              xe2x80x83                                                              ⁢                                          xe2x80x83                                        ⁢                                                                  x                        i                                            ⁢                                              xe2x80x83                                            ⁢                                              y                        i                                                                                                                  ]                                              (        3.19        )            
That is, a linear phase can be estimated from the following equations (3.20.1) and (3.20.2).                     a        =                                                            ∑                                  xe2x80x83                                            ⁢                              xe2x80x83                            ⁢                                                x                  i                  2                                ⁢                                  xe2x80x83                                ⁢                                                      ∑                                          xe2x80x83                                                        ⁢                                      xe2x80x83                                    ⁢                                      y                    i                                                                        -                                          ∑                                  xe2x80x83                                            ⁢                              xe2x80x83                            ⁢                                                x                  i                                ⁢                                  xe2x80x83                                ⁢                                                      ∑                                          xe2x80x83                                                        ⁢                                      xe2x80x83                                    ⁢                                                            x                      i                                        ⁢                                          xe2x80x83                                        ⁢                                          y                      i                                                                                                                              L              ⁢                              xe2x80x83                            ⁢                                                ∑                                      xe2x80x83                                                  ⁢                                  xe2x80x83                                ⁢                                  x                  i                  2                                                      -                                          (                                                      ∑                                          xe2x80x83                                                        ⁢                                      xe2x80x83                                    ⁢                                      x                    i                                                  )                            2                                                          (                  3.20          ⁢          .1                )                                b        =                                            L              ⁢                              xe2x80x83                            ⁢                                                ∑                                      xe2x80x83                                                  ⁢                                  xe2x80x83                                ⁢                                                      x                    i                                    ⁢                                      xe2x80x83                                    ⁢                                      y                    i                                                                        -                                          ∑                                  xe2x80x83                                            ⁢                              xe2x80x83                            ⁢                                                x                  i                                ⁢                                  xe2x80x83                                ⁢                                                      ∑                                          xe2x80x83                                                        ⁢                                      xe2x80x83                                    ⁢                                      y                    i                                                                                                          L              ⁢                              xe2x80x83                            ⁢                                                ∑                                      xe2x80x83                                                  ⁢                                  xe2x80x83                                ⁢                                  x                  i                  2                                                      -                                          (                                                      ∑                                          xe2x80x83                                                        ⁢                                      xe2x80x83                                    ⁢                                      x                    i                                                  )                            2                                                          (                  3.20          ⁢          .2                )            
In the present invention, when a linear phase is estimated from a continuous phase, a linear trend estimating method is used.
As apparent from the above discussion, in the conventional method of measuring a jitter, a peak-to-peak jitter is measured in time domain using an oscilloscope and an RMS jitter is measured in frequency domain using a spectrum analyzer.
In the method of measuring a jitter in time domain, a peak-to-peak jitter JPP of a clock signal is measured in time domain. A relative fluctuation between zero crossings becomes a problem as a peak-to-peak jitter JPP. Therefore, for example, like a clock signal in a computer or the like shown in FIG. 81a, when a jitter-free clock signal has a waveform shown by a dotted line, in a jittery clock signal, for example paying attention to a rising point of the waveform, a time interval Tint between a rising point and a next rising point each fluctuating toward leading side and trailing side from the rising point of the dotted line waveform as a center fluctuates. This instantaneous interval Tint is obtained as a peak-to-jitter JPP. FIGS. 23 and 24 show a measured example of a peak-to-peak jitter measured using an oscilloscope and the measuring system, respectively. A clock signal under test is applied to a reference input of the phase detector. In this case, the phase detector and the signal generator compose a phase-locked loop. A signal of the signal generator is synchronized with the clock signal under test and is supplied to an oscilloscope as a trigger signal. In this example, a jitter of rise edge of the clock signal is observed. A square zone is used to specify a level to be crossed by the signal. A jitter is measured as a varying component of time difference between xe2x80x9ca time point when the clock signal under test crosses the specified levelxe2x80x9d and xe2x80x9ca reference time point given by the trigger signalxe2x80x9d. This method.requires a longer time period for the measurement. For this reason, the trigger signal must be phase-synchronized with the clock signal under test so that the measurement is not influenced by a frequency drift of the clock signal under test.
A measurement of a jitter in time domain corresponds to a measurement of a fluctuation of a time point when a level is crossed by the signal. This is called, in the present invention, a zero crossing method. Since a change rate of a waveform is maximum at the zero crossing, a timing error of a time point measurement is minimum at the zero crossing.                               Δ          ⁢                      xe2x80x83                    ⁢          t                =                              "LeftBracketingBar"                                          Δ                ⁢                                  xe2x80x83                                ⁢                A                                            A2                ⁢                                  xe2x80x83                                ⁢                π                ⁢                                  xe2x80x83                                ⁢                                  f                  0                                ⁢                                  xe2x80x83                                ⁢                sin                ⁢                                  xe2x80x83                                ⁢                                  (                                      2                    ⁢                                          xe2x80x83                                        ⁢                    π                    ⁢                                          xe2x80x83                                        ⁢                                          f                      0                                        ⁢                                          xe2x80x83                                        ⁢                    t                                    )                                                      "RightBracketingBar"                    ≥                                    Δ              ⁢                              xe2x80x83                            ⁢              A                                      2              ⁢                              xe2x80x83                            ⁢              π              ⁢                              xe2x80x83                            ⁢                              f                0                            ⁢                              xe2x80x83                            ⁢              A                                                          (        3.21        )            
In FIG. 25(a), the zero crossing is indicated by each of small circles. A time interval between a time point ti that a rise edge crosses a zero amplitude level and a.time point ti+2 that a next rise edge crosses a zero amplitude level gives a period of this cosine wave. FIG. 25(b) shows an instantaneous period Pinst obtained from the zero crossing (found from adjacent zero crossings tj+1 and ti+2). A instantaneous frequency finst is given by an inverse number of Pinst.
Pinst(ti+2)=ti+2xe2x88x92ti,pinst(ti+2)=2(ti+2xe2x88x92ti+1)xe2x80x83xe2x80x83(3.22.1)
xe2x80x83finst(ti+2)=1/pinst(ti+2)xe2x80x83xe2x80x83(3.22.2)
Problems in measuring a jitter in time domain will be discussed. In order to measure a jitter, a rise edge of a clock signal under test XC(t) is captured, using an oscilloscope, at a timing of the zero crossing.
Xc(t)=Ac cos(2xcfx80fct+xcex8c+xcex94xcfx86(t))xe2x80x83xe2x80x83(3.23)
This means that only XC(t) satisfying the next condition of phase angle given by the following equation (3.24) can be collected.                                           2            ⁢                          xe2x80x83                        ⁢            π            ⁢                          xe2x80x83                        ⁢                          f              0                        ⁢                          xe2x80x83                        ⁢                          t                              3                ⁢                                  xe2x80x83                                ⁢                                  π                  /                  2                                                              +                      θ            c                    +                      Δ            ⁢                          xe2x80x83                        ⁢            φ            ⁢                          xe2x80x83                        ⁢                          (                              t                                  3                  ⁢                                      xe2x80x83                                    ⁢                                      π                    /                    2                                                              )                                      =                                            ±              2                        ⁢                          xe2x80x83                        ⁢            m            ⁢                          xe2x80x83                        ⁢            π                    +                                    3              ⁢                              xe2x80x83                            ⁢              π                        2                                              (        3.24        )            
A probability density function of a sample corresponding to the zero crossing of a rise edge is given by the following equation (3.25). (For example, refer to a reference literature c10.)                                           1                          2              ⁢                              xe2x80x83                            ⁢              π              ⁢                                                                    A                    c                    2                                    -                                                            x                      c                      2                                        ⁢                                          xe2x80x83                                        ⁢                                          (                      t                      )                                                                                                    "RightBracketingBar"                                                    x              c                        ⁢                          xe2x80x83                        ⁢                          (              t              )                                =          0                                    (        3.25        )            
Therefore, a time duration required for randomly sampling a clock signal under test to collect phase noises xcex94xcfx86(t3xcfx80/2) of N points is given by the following equation (3.26).
(2xcfx80Ac)(NT0)xe2x80x83xe2x80x83(3.26)
That is, since only zero crossing samples can be utilized for a jitter estimation, at least (2xcfx80Ac) times of test time period is required compared with an usual measurement.
As shown in FIG. 26, the magnitude of a set of phase noises which can be sampled by the zero crossing method is smaller than an entire set of phase noises. Therefore, a peak-to-peak jitter       J    pp    ,      3    ⁢          xe2x80x83        ⁢          π      /      2      
which can be estimated is equal to or smaller than a true peak-to-peak jitter JPP.
Jpp=maxk(xcex94xcfx86(k))xe2x88x92minl(xcex94xcfx86(l))                               J          pp                ,                              3            ⁢                          xe2x80x83                        ⁢                          π              /              2                                ≤                      J            pp                                              (        3.27        )            
The worst drawback of the zero crossing method is that a time resolution of the period measurement cannot be selected independently on a period of a signal under test. The time resolution of this method is determined by a period of the signal under test, i.e., the zero crossing. FIG. 27 is a diagram in which the zero crossings of the rise edges are plotted on a complex plane. The sample in the zero crossing method is only one point indicated by an arrow, and the number of samples per period cannot be increased. When a number nj is given to the zero crossing of a rise edge, the zero crossing method measures a phase difference expressed by the following equation (3.28).
ni(2xcfx80)xe2x80x83xe2x80x83(3.28)
As a result, an instantaneous period measured by the zero crossing method comes to, as shown in FIG. 25(b), a rough approximation obtained by use of a step function.
In 1988, David Chu invented a time interval analyzer (for example, refer to reference literatures c12 and c13). In the time interval analyzer, when integer values ni of the zero crossings ni(2xcfx80) of the signal under test are counted, the elapsed time periods ti are also simultaneously counted. By this method, the time variation of the zero crossing with respect to the elapsed time period could be plotted. Further, by using (ti, ni), a point between measured data can smoothly be interpolated by spline functions. As a result, it was made possible to observe an instantaneous period approximated in higher order. However, it should be noted that David Chu""s time interval analyzer is also based on the zero crossing measurement of a signal under test. Although the interpolation by spline functions makes it easier to understand the physical meaning, the fact is that only the degree of approximation of an instantaneous period is increased. Because, the data existing between the zero crossings have not been still measured. That is, the time interval analyzer cannot either exceed the limit of the zero crossing method. An opposite example for interpolating the instantaneous data will be discussed later.
Next, a method of measuring a jitter in frequency domain will be described.
An RMS jitter JRMS of a clock signal is measured in frequency domain. For example, in a data communication, a deviation from an ideal timing point becomes a problem as an RMS jitter JRMS. Therefore, as shown in FIG. 81b, when a jitter-free square wave signal has a waveform shown by a dotted line, a rising timing of a jittery waveform fluctuates. In this case, a deviating width of an actual rising point (solid line) against a normal rising point (dotted line) is obtained as an RMS jitter JRMS. FIGS. 28 and 29 show an example of an RMS jitter measured by using a spectrum analyzer and a measuring system using a spectrum analyzer, respectively. A clock signal under test is inputted to a phase detector as a reference frequency. In this case, the phase detector and the signal generator compose a phase-locked loop. A phase difference signal between the clock signal under test detected by the phase detector and the signal from the signal generator is inputted to the spectrum analyzer to observe a phase noise spectrum density function. An area below the phase noise spectrum curve shown in FIG. 28 corresponds to an RMS jitter JRMS. The frequency axis expresses the offset frequencies from the clock frequency. That is, zero (0) Hz corresponds to the clock frequency.
A phase difference signal xcex94xcfx86(t) between the clock signal under test XC(t) expressed by the equation (3.23) and a reference signal expressed by the following equation (3.29) is outputted from the phase detector.
xref(t)=A cos(2xcfx80fct+xcex80)xe2x80x83xe2x80x83(3.29)
At this point in time, since the reference signal being applied to a phase-locked loop circuit (PLL circuit) under test has a constant period, the phase difference signal xcex94xcfx86(t) corresponds to a phase noise waveform. When the phase difference signal xcex94xcfx86(t) observed during a finite time period T and is transformed into frequency domain, a phase noise power spectrum density function Gxcex94xcfx86xcex94xcfx86(f) can be obtained.                                           S                          Δ              ⁢                              xe2x80x83                            ⁢              φ                                ⁢                      xe2x80x83                    ⁢                      (            f            )                          =                              ∫            0            T                    ⁢                      Δ            ⁢                          xe2x80x83                        ⁢            φ            ⁢                          xe2x80x83                        ⁢                          (              t              )                        ⁢                          xe2x80x83                        ⁢                          ⅇ                                                -                  2                                ⁢                                  xe2x80x83                                ⁢                π                ⁢                                  xe2x80x83                                ⁢                ft                                      ⁢                          xe2x80x83                        ⁢                          ⅆ              t                                                          (        3.30        )                                                      G                          Δ              ⁢                              xe2x80x83                            ⁢              φ              ⁢                              xe2x80x83                            ⁢              Δ              ⁢                              xe2x80x83                            ⁢              φ                                ⁢                      xe2x80x83                    ⁢                      (            f            )                          =                              lim                          T              →              ∞                                ⁢                      xe2x80x83                    ⁢                                    2              T                        ⁢                          xe2x80x83                        ⁢                          E              ⁡                              [                                                      "LeftBracketingBar"                                                                  S                                                  Δ                          ⁢                                                      xe2x80x83                                                    ⁢                          φ                                                                    ⁢                                              xe2x80x83                                            ⁢                                              (                        f                        )                                                              "RightBracketingBar"                                    2                                ]                                                                        (        3.31        )            
From Parseval""s theorem, a mean square value of a phase noise waveform is given by the following equation (3.32). (For example, refer to a reference literature c14.)                                           E            ⁡                          [                              Δ                ⁢                                  xe2x80x83                                ⁢                                                      φ                    2                                    ⁡                                      (                    t                    )                                                              ]                                ≡                                    lim                              T                →                ∞                                      ⁢                                          1                T                            ⁢                                                ∫                  0                  T                                ⁢                                                                            Δφ                      2                                        ⁡                                          (                      t                      )                                                        ⁢                                      xe2x80x83                                    ⁢                                      ⅆ                    t                                                                                      =                              ∫            0            ∞                    ⁢                                                    G                ΔφΔφ                            ⁡                              (                f                )                                      ⁢                          xe2x80x83                        ⁢                          ⅆ              f                                                          (        3.32        )            
That is, it can be understood that by measuring a sum of the power spectrum, a mean square value of a phase noise waveform can be estimated. A positive square root of the mean square value (an effective value) is called RMS (a root-mean-square) jitter JRMS.                               J          RMS                =                                            ∫              0                              f                MAX                                      ⁢                                          G                                  Δ                  ⁢                                      xe2x80x83                                    ⁢                  φ                  ⁢                                      xe2x80x83                                    ⁢                  Δ                  ⁢                                      xe2x80x83                                    ⁢                  φ                                            ⁢                              xe2x80x83                            ⁢                              (                f                )                            ⁢                              xe2x80x83                            ⁢                              ⅆ                f                                                                        (        3.33        )            
When a mean value is zero, a mean square value is equivalent to a variance, and an RMS jitter is equal to a standard deviation.
As shown in FIG. 28, JRMS can be accurately approximated to a sum of Gxcex94xcfx86xcex94xcfx86(f) in the proximity of the clock frequency (for example, refer to a reference literature c15). Actually, in the equation (3.33), the upper limit value fMAX of the frequency of Gxcex94xcfx86xcex94xcfx86(f) to be summed is (2fcxe2x88x92xcex5). Because, if Gxcex94xcfx86xcex94xcfx86 (f) is summed in the frequency range wider than the clock frequency, the harmonics of the clock frequency are included in JRMS.
In a measurement of an RMS jitter in frequency domain, there are required a phase detector, a signal generator whose phase noise is small and a spectrum analyzer. As can be understood from the equation (3.33) and FIG. 28, a phase noise spectrum is measured by frequency-sweeping a low frequency range. For this reason, the measuring method requires a measurement time period of approximately 10 minutes, and cannot be applied to the test of a microprocessor. In addition, in the measurement of an RMS jitter in frequency domain, a peak-to-peak jitter cannot be estimated since the phase information has been lost.
As described above, in the conventional method of measuring a jitter, a peak-to-peak jitter is measured in time domain using an oscilloscope. The basic method of measuring a jitter in time domain is the zero crossing method. The biggest drawback of this method is that a time resolution of a period measurement cannot be made fine independently on the period of a signal under test. For this reason, a time interval analyzer for simultaneously counting the integer values ni of the zero crossings of the signal under test ni (2xcfx80) and the elapsed time periods ti was invented. However, the data existing between the zero crossings cannot be measured. That is, the time interval analyzer also cannot exceed the limit of the zero crossing method.
On the other hand, an RMS jitter is measured in frequency domain using a spectrum analyzer. Since the phase information has been lost, a peak-to-peak jitter cannot be estimated.
In addition, either case of measuring a jitter in time domain or measuring an RMS jitter in frequency domain requires a measurement time of approximately 10 minutes. In a test of a VLSI, a testing time of only approximately 100 msec is allocated to one test item. Therefore, there is a serious drawback in the conventional method of measuring a jitter that the method cannot be applied to a test of a VLSI in the manufacturing process thereof.
A clock frequency of a microcomputer has been shifting toward a higher frequency with a rate of 2.5 times per five years. Therefore, a clock jitter of a microcomputer cannot be measured unless the method of measuring a clock jitter is scalable with respect to the measuring time resolution. A peak-to-peak jitter has conventionally been measured in time domain using a oscilloscope or a time interval analyzer. In order to measure a peak-to-peak jitter of a clock signal having a higher frequency using those measuring devices, it is necessary to increase a sampling rate (the number of samples per second) or to decrease a sampling interval. That is, those hardware devices must be developed at least every five years.
Problems in measuring a jitter of CD or DVD will be described. In CD or DVD, a light beam is focused on a disk, and a reflected light returned from a pit is detected by an optical pick-up, and then the detected light is converted into an RF signal (an electrical signal) by a photo-diode. The pit on the disk is formed in the state that the pit is slightly elongated or shortened in its length direction. As a result, rising-up and falling-down characteristics (duty ratio) of the RF signal become asymmetric. For example, when an eye pattern of the RF signal is observed using an oscilloscope, its center is shifted along the y axis. Therefore, in order to evaluate a jitter of the disk, the rising edge and the falling edge of the RF signal must be distinguished. In the measurement of an RMS jitter using a spectrum analyzer, the rising edge and the falling edge of the RF signal cannot be distinguished.
In addition, as mentioned above, a clock frequency of a microcomputer has been increased with a rate of 2.5 times per five years. For measuring a peak-to-peak jitter of a clock signal having a higher frequency, it is necessary for an AD converter for inputting to a digital oscilloscope to operate at higher speed in accordance with the higher frequency of the clock signal and to have a resolution equal to or more than eight bits.
It is an object of the present invention to provide an apparatus for and a method of measuring a jitter wherein a peak-to-peak jitter or an RMS jitter can be measured in a short test time of approximately 100 msec or so.
It is another object of the present invention to provide an apparatus for and a method of measuring a jitter wherein data obtained from the conventional RMS jitter measurement or the conventional peak-to-peak jitter measurement can be utilized.
It is further another object of the present invention to provide a scalable apparatus for and a scalable method of measuring a jitter.
It is yet further another object of the present invention to provide an apparatus for and a method of measuring a jitter that can measure a peak-to peak jitter and/or an RMS jitter each corresponding to a rising edge or a falling edge of a waveform.
It is yet further another object of the present invention to provide an apparatus for measuring a jitter that does not require an AD converter.
It is yet further another object of the present invention to provide an apparatus for measuring a jitter that is compatible with a method of measuring a peak-to-peak jitter of a conventional zero crossing method shown in FIG. 24 and/or a method of measuring an RMS jitter of a phase detecting method shown in FIG. 29.
It is yet further another object of the present invention to provide an apparatus that can measure a cycle-to-cycle jitter.
It is yet further another object of the present invention to provide an apparatus for measuring a jitter that can measure a histogram of jitter.
In order to achieve the above objects, in one aspect of the present invention, there is provided an apparatus for measuring a jitter wherein a clock waveform XC(t) is transformed into a complex analytic signal using analytic signal transforming means to obtain, by linear phase eliminating means, a variable term that is obtained by eliminating linear phase from an instantaneous phase of this analytic signal, i.e., a phase noise waveform xcex94xcfx86(t), and a jitter of the clock waveform is obtained, by jitter detecting means, from this phase noise waveform.
In another aspect of the present invention, there is provided a method of measuring a jitter comprising the steps of: transforming a clock waveform XC(t) into a complex analytic signal; estimating a variable term that is obtained by removing a linear phase from an instantaneous phase of this analytic signal, i.e., a phase noise waveform xcex94xcfx86(t); and obtaining a jitter from the phase noise waveform.
An RMS jitter is obtained from the phase noise waveform. In addition, the phase noise waveform is sampled in the proximity of a zero crossing point of real part of an analytic signal, and a differential waveform of the sample phase noise waveform is calculated to obtain a peak-to-peak jitter from the differential phase noise waveform.
There are provided a scalable apparatus for and a scalable method of measuring a jitter that are constructed such that the clock waveform is frequency-divided by a frequency divider, and thereafter the frequency-divided clock waveform is transformed into an analytic signal.
In further another aspect of the present invention, the clock waveform is compared with a reference analog quantity by a comparator, and an output signal of the comparator is transformed into an analytic signal.